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Current and Future Directions in Applied Mathematics pp 219–261Cite as

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Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

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Abstract

This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a discretization technique of Veselov.

The general idea for these variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators.

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Author information

Authors and Affiliations

  1. Control and Dynamical Systems 116-81, California Institute of Technology, Pasadena, CA, 91125, USA

    Jerrold E. Marsden

  2. Mechanical Engineering, University of California at Berkeley, Berkeley, CA, 94720, USA

    Jeffrey M. Wendlandt

Authors
  1. Jerrold E. Marsden
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  2. Jeffrey M. Wendlandt
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Notre Dame, 46556-5683, Notre Dame, IN, USA

    Mark Alber , Bei Hu  & Joachim Rosenthal ,  & 

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Marsden, J.E., Wendlandt, J.M. (1997). Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms. In: Alber, M., Hu, B., Rosenthal, J. (eds) Current and Future Directions in Applied Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-2012-1_18

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  • DOI: https://doi.org/10.1007/978-1-4612-2012-1_18

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-7380-6

  • Online ISBN: 978-1-4612-2012-1

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